convergence of a boundary integral method for 3-d...

DISCRETE AND CONTINUOUS Website: http://AIMsciences.orgDYNAMICAL SYSTEMS–SERIES BVolume 2, Number 1, February 2002 pp. 1–34

CONVERGENCE OF A BOUNDARY INTEGRAL METHODFOR 3-D WATER WAVES

Thomas Y. Hou

Applied Mathematics

California Institute of TechnologyPasadena, CA 91125

Pingwen Zhang

School of Mathematical SciencesPeking University

Beijing 100871, China

Abstract. We prove convergence of a modified point vortex method for time-

dependent water waves in a three-dimensional, inviscid, irrotational and incompress-ible fluid. Our stability analysis has two important ingredients. First we derive aleading order approximation of the singular velocity integral. This leading orderapproximation captures all the leading order contributions of the original velocity

integral to linear stability. Moreover, the leading order approximation can be ex-pressed in terms of the Riesz transform, and can be approximated with spectralaccuracy. Using this leading order approximation, we construct a near field cor-rection to stabilize the point vortex method approximation. With the near field

correction, our modified point vortex method is linearly stable and preserves all thespectral properties of the continuous velocity integral to the leading order. Nonlin-ear stability and convergence with 3rd order accuracy are obtained using Strang’stechnique by establishing an error expansion in the consistency error.

1. Introduction. We prove convergence of a modified point vortex method fortime-dependent water waves in a three-dimensional, inviscid, irrotational and in-compressible fluid. Boundary integral methods have been one of the commonlyused numerical methods in studying fluid dynamical instabilities associated withfree interface problems. They have the advantage of reducing the problem of thefree surface flow to one defined on the interface only and thus allow higher orderapproximations of the interface. Computations using a boundary integral formu-lation in three space dimensions include [2, 6, 14, 20, 21, 25]. On the other hand,boundary integral methods often suffer from high frequency numerical instabilities[22, 11, 15]. Various stabilizing methods have been proposed in the literature toalleviate this difficulty [11, 3, 5]. Using a spectral discretization and certain Fourierfiltering, Beale, Hou and Lowengrub [5] proved convergence of a spectrally accurateboundary integral method for two-dimensional water waves. A key idea is to en-force a compatibility between the quadrature rule of the singular velocity integraland that of the spatial derivative. This discrete compatibility is achieved by usinga Fourier filtering. The amount of filtering is determined by the quadrature rule inapproximating the velocity integral and the derivative rule being used.

1991 Mathematics Subject Classification. Primary; 65M12; Secondary, 76B15.

Key words and phrases. Water waves, boundary integral methods, stability.

1

2 THOMAS Y. HOU AND PINGWEN ZHANG

The stability of boundary integral methods for three-dimensional water waves isconsiderably more difficult than the corresponding two-dimensional problem. Thetwo-dimensional interface problem has a special property, namely the singular ker-nel has a removable simple pole singularity. As a consequence, spectrally accurateapproximations can be constructed for the velocity integral and certain compatibil-ity between the quadrature for velocity integral and the quadrature for the deriva-tive can be enforced. In a three-dimensional free surface, the singular kernel has abranch point singularity which is not removable. Straightforward approximationsof the singular integrals do not preserve the spectral properties of these singularintegral operators. By the spectral properties of a singular integral operator, wemean the high frequency information contained in the Fourier transform of a sin-gular integral operator. The well-posedness of 3-D water waves is a consequence ofthe subtle balance of the spectral properties of various singular integral operators.Violation of this subtle balance makes it susceptible to numerical instability. Thisis why it is difficult to design a numerically stable boundary integral method forthree-dimensional water waves.

In this paper, we propose a stable and convergent boundary integral methodfor three-dimensional water waves. The method is a variant of the point vortexapproximation. Without any modification, the point vortex method approximationis numerically unstable for three-dimensional water waves. Our stability analysishas two important ingredients. First we derive a leading order approximation of thesingular velocity integral. This leading order approximation captures all the leadingorder contributions of the original velocity integral to linear stability. Moreover,the leading order approximation can be expressed in terms of the Riesz transform,and can be approximated with spectral accuracy. Thus the spectral propertiesof the original velocity integral are preserved exactly at the discrete level by thisleading order approximation. Secondly, the term consisting of the difference ofthe original velocity integral and the leading order approximation is less singularand does not contribute to linear stability. Thus it can be approximated by anyconsistent quadrature rule such as the point vortex method approximation. Sincethe leading order approximation in effect has desingularized the velocity integral,we obtain an improved 3rd order accuracy of the (modified) point vortex methodapproximation for 3-D water waves.

Our boundary integral method can be interpreted as a stabilizing method forthe point vortex method. The difference between the spectral discretization andthe point vortex discretization of the leading order approximation constitutes anear field correction to the original point vortex method. This nonlocal correctionaccounts for the near field contribution of the singular velocity integral. Althoughthis correction term is small (of order O(h)), it contains the critical small scale in-formation which is essential to the stability of the boundary integral method. Withthe near field correction, we can show that our modified point vortex method pre-serves all the spectral properties of the linearized operators at the continuum level.As a consequence, we prove that the modified point vortex method is stable andconvergent with third order accuracy. Furthermore, using a generalized arclengthframe (see section 6, and [16]), the near field correction becomes a convolution oper-ator which can be computed by Fast Fourier Transform with O(M logM) operationcount, here M = N2 is the total number of discrete Lagrangian particles on thefree surface.

CONVERGENCE OF A BOUNDARY INTEGRAL METHOD FOR 3-D WATER WAVES 3

As in the convergence study for a boundary integral method for 2-D water wavesin [5], it is important to separate the treatment of linear stability from that ofnonlinear stability. It is linear stability that plays the most important role inobtaining convergence of a boundary integral method, as long as the solution issufficiently smooth. As observed by Strang [24], if there exists an error expansionfor the consistency error and the problem is linearly stable, then nonlinear stabilitycan be obtained by the smallness of the error. In conventional convergence analysis,we usually compare our numerical solution with the exact solution. Strang’s trick isnot to use the “exact” solution in studying convergence, but to construct a “smoothapproximate solution” that is O(hp) perturbation of the “exact” solution (p is theorder of the numerical scheme being considered). This smooth approximate solutionsatisfies the discrete equations more accurately: R(t) = O(hr) for arbitrarily large ras long as the continuous solution is sufficiently smooth. Existence of such smoothparticles is guaranteed by the existence of the error expansion for the consistencyerror. Then, in the stability step, one can bound e(t), the difference between thesmooth approximate solution and the numerical solution. If the numerical methodis linearly stable, nonlinear stability can be obtained by using the smallness of theerror e(t). This greatly simplifies the nonlinear stability analysis. In our modifiedpoint vortex method, the order of accuracy is p = 3. This is not accurate enoughand Strang’s trick has to be applied to obtain nonlinear stability. In section 5, weprove the existence of the error expansion in terms of the odd powers of h. Wethen use this error expansion to construct a smooth particle solution which satisfiesthe modified point vortex method with 5th order accuracy. Thus application ofStrang’s technique proves nonlinear stability and convergence.

We remark that Beale [4] has recently analyzed convergence of a boundary in-tegral method for three-dimensional water waves. The method analyzed in [4] isbased on a different boundary formulation and uses a desingularization in the inte-gral formulation. Another important ingredient in his analysis is the use of a specialcut-off function in regularizing the singular kernel. One advantage of using desingu-larization in the boundary integral formulation is that it does not require smoothingto control the aliasing error. In this paper, we also use a similar desingularizationtechnique to avoid the need of using Fourier smoothing.

The organization of the rest of the paper is as follows. In section 2, we presenta boundary integral formulation for three-dimensional water waves. In section 3,we demonstrate the instability of the classical point vortex method. In section 4,we introduce our new stabilizing technique and present our modified point vortexmethod. Section 5 is devoted to the consistency analysis. Section 6 is devoted tostudying properties of some singular integral operators which are closely relatedto our stability analysis. In section 7, we present our stability and convergenceanalysis. The proof of several technical lemmas is deferred to the Appendix.

2. A Boundary Integral Formulation for 3-D Water Waves. We first reviewthe boundary integral formulation for the 3-D water wave problem. Throughoutthe paper, we will use bold face letters to denote vector variables. We assumethat the flow is inviscid, incompressible, irrotational, and is separated by a freeinterface. We assume that the fluid has infinite depth. The state of the system at atime t is specified by the interface x(α, t) and the velocity potential φ(α, t) on theinterface, where α = (α1, α2). To simplify the notations, we often drop the timevariable from now on, but all the quantities, x, φ and µ will be time-dependent. To


express the evolution, we need to write the velocity on the surface in terms of thesevariables. Following [1, 14], we begin with a double layer or dipole representationfor the potential in terms of the dipole strength µ(α), to be determined from φ. Wewrite the potential in the fluid domain as

φ(x) =∫ ∫

µ(α′)N(α′) · ∇x′G(x− x(α′))dα′, (1)

where N(α′) = xα1(α′)×xα2(α

′) is the unnormalized outward normal to the surface,

G(x− x′) = − 14π|x− x′|

,

is the free space Green function for the Laplace equation, and

∇x′G(x− x′) = − x− x′

4π|x− x′|3.

We define the corresponding unit outward normal vector as n(α) = N(α)/|N(α)|.It follows from the properties of the double layer potential that the value of φ onthe interface is given by

φ(α) =12µ(α) +Kµ(α), (2)

where φ(α) = φ(x(α)) and

Kµ(α) =∫ ∫

µ(α′)N(α′) · ∇x′G(x(α)− x(α′))dα′. (3)

Differentiating both sides of equation (1) with respect to x and integrating by parts,we obtain

∇φ(x) =∫ ∫

η(α′)×∇x′G(x− x(α′))dα′, (4)

where η(α) = (γ1xα2 − γ2xα1)(α), and γi = ∂µ∂αi

, i = 1, 2. Since the tangentialvelocity of the interface is not unique, we need to specify an interface velocity. Forwater waves, it is customary to evolve the interface with the interface velocity fromthe fluid domain. Thus, to obtain the interface velocity, we need to compute thelimiting value of ∇φ(x) as x approaches to the interface from below. By combin-ing the stream function formulation with the double layer potential formulation,Haroldsen and Meiron in [14] have shown that the interface velocity is given by

w(α, t) =∫ ∫

η(α′)×∇x′G(x(α)− x(α′))dα′ +12η(α)× xα1 × xα2

|xα1 × xα2 |2. (5)

To make it easier for our presentation, we denote by w0 the integral part of theinterface velocity,

w0(α) =∫ ∫

η(α′)×∇x′G(x(α)− x(α′))dα′, (6)

and by wloc the local velocity field

wloc(α) =12η(α)× xα1 × xα2

|xα1 × xα2 |2. (7)

In Lagrangian formulation, we evolve the free surface by the interface velocityderived above, i.e.

∂x∂t

(α, t) = w(α, t), x(α, 0) = x0(α). (8)


For the evolution of φ(α, t), we use Bernoulli’s equation. If we neglect surfacetension, Bernoulli’s equation in the Lagrangian frame is

φt −12|w|2 + g · x = 0, (9)

where g = (0, 0, g), g is the gravity acceleration coefficient. The evolution equations(8) and (9), together with the relations (2)-(5) completely specify the motion of thesystem.

Using equation (2) directly in numerical approximations is more sensitive tonumerical instability. To alleviate this difficulty, we first derive an equivalent for-mulation for equation (2) which gives better numerical stability property than (2).To this end, we differentiate equation (2) with respect to αl. Using (4)-(5), weobtain

φαl=

γl

2+ xαl

·w0, l = 1, 2. (10)

Note that γ1 and γ2 are not independent. Thus equation (10) as it stands is notinvertible in general. To solve for γ1 and γ2, one has to supplement equation (10)by the constraint γl = µαl

, l = 1, 2. Instead of solving (10) subject to the aboveconstraint, we will use an equivalent formulation of (10) for γ in our numericaldiscretization:

φαl=γl

2+ Pl1(xα1 ·w0) + Pl2(xα2 ·w0), l = 1, 2. (11)

where Pij (i, j = 1, 2) are projection operators with zero constant mode. Moreprecisely we define Pij in Fourier transform space as follows: (Pij)k = kikj

|k|2 for

k 6= 0, where k = (k1, k2), and (Pij)k = 0 for k = 0. Here (Pij)k stands for theFourier transform of Pij . Using the definition of operator Pij , we have

D2P11 = D1P12, D1P22 = D2P12, P12 = P21, P11 + P22 = I, (12)

for functions with zero modes, i.e. f0 = 0. Here Dl = ∂∂αl

is a partial derivativeoperator with respect to ∂αl, l = 1, 2. The reason that equation (11) has a betterstability property than equation (2) at the discrete level is because in derivingequation (11) we have performed one integration by parts to cancel the leadingorder singular terms.

We now show that (2) and (11) are equivalent formulations at the continuumlevel. We assume that φ0 = µ0/2. As we mentioned earlier, γ1 and γ2 are notindependent. Using (11) and (12), we can prove (γ1)α2 = (γ2)α1 . Thus there existsµ, such that γl = µαl

, l = 1, 2. To solve for γl from (11), we differentiate (11) withrespect to αl and add the resulting equations. This gives

4φ =4µ2

+ (xα1 ·w0)α1 + (xα2 ·w0)α2 , (13)

where we have used the identities:

D1P11 +D2P21 = D1P11 +D1P22 = D1(P11 + P22) = D1,

D1P12 +D2P22 = D2P11 +D2P22 = D2(P11 + P22) = D2,

which follows from (12).Define 4−1 as follows:(

4−1)k

= − 1|k|2

, if k 6= 0,(4−1

)0

= 0.


Applying the 4−1 operator to the both sides of the above equation, we get theequation for µ

φ =µ

2+4−1 ((xα1 ·w0)α1 + (xα2 ·w0)α2) , (14)

where we have used the fact that φ0 = µ0/2. Using integration by parts andequation (4), we can show that ∂K

∂αj= xαj ·w0, which implies

(xα1 ·w0)α1 + (xα2 ·w0)α2 = 4K(µ).

Therefore, equation (11) is equivalent to equation (2) up to a constant. Sinceφ0 = µ0/2, the constant must be zero. Thus the solvability of (2) for µ [1] impliessolvability of (11) for γ.

Next we express (11) as follows:

(12I +A)γ = 5αφ, (15)

where A is two by two matrix, A = (aij), with

a11 = P11K1 + P12K3, a12 = P11K2 + P12K4,

a21 = P21K1 + P22K3, a22 = P21K2 + P22K4,

and

K1f(α) = xα1(α) ·∫f(α′)xα2(α

′)×5x′G(x(α)− x(α′))dα′,

K2f(α) = −xα1(α) ·∫f(α′)xα1(α

′)×5x′G(x(α)− x(α′))dα′,

K3f(α) = xα2(α) ·∫f(α′)xα2(α

′)×5x′G(x(α)− x(α′))dα′,

K4f(α) = −xα2(α) ·∫f(α′)xα1(α

′)×5x′G(x(α)− x(α′))dα′.

Recall that an integral kernel K(x, y) is called weakly singular (see e.g. page6 of [7]) if there exists a positive constant M and α ∈ (0, 2] such that for allx, y ∈ G, x 6= y, we have

|K(x, y)| ≤M |x− y|α−2.

It is easy to prove that the kernels Ki, i = 1, 2, 3, 4 defined above are integraloperators with weakly singular kernels. It follows from Theorem 1.11 in page 6 of [7]that Ki, i = 1, 2, 3, 4 are compact operators. Moreover, PijKl is a compact operatorsince Pij is a bounded operator (see Definition 1.1 in page 2 of [7]). ConsequentlyA is a compact operator (see Theorem 1.4 in page 2 of [7]).

By the solvability of (11) for γ, we have

(12I +A)γ = 0 =⇒ γ = 0. (16)

Using (16) and the fact that A is a compact operator, we can apply Theorem 1.16in Colton and Kress [7] (page 13) to show that (1

2I +A)−1 exists and is bounded.Moreover, we define a set of complementary tangent vectors x∗αl

and x∗α2as

follows:

x∗α1=

1|xα1 × xα2 |

(xα2 × n), x∗α2=

1|xα1 × xα2 |

(n× xα1).


It is easy to verify that

x∗αl· xαk

= δlk, l, k = 1, 2, (17)

where δlk are the Kronecker delta functions. By projecting the interface velocityw(α) into x∗α1

, x∗α2and n vectors respectively, we obtain using (17) that

w(α) = w0 + wloc

= (w · xα1)x∗α1

+ (w · xα2)x∗α2

+ (w · n)n= (φα1)x

∗α1

+ (φα2)x∗α2

+ (w0 · n)n , (18)

where we have used

d

dαlφ(x(α)) = ∇φ · xαl

= w · xαl, l = 1, 2.

In summary, the evolution equations for the 3-D water wave problem are as follows:

xt = w(α) = (φα1)x∗α1

+ (φα2)x∗α2

+ (w0 · n)n, (19)

φt =12|w(α)|2 − g · x, (20)

φαl=

γl

2+ Pl1(xα1 ·w0) + Pl2(xα2 ·w0), l = 1, 2, (21)

where w0 is defined in (6).From now on, with x(α, t) = (α, 0) + s(α, t), we assume that s(α, t) and φ(α, t)

are double periodic in α with period 2π. To reduce the computational domain toa single period, we need to replace the original Green’s function by a periodic one,which is obtained by summing up all its periodic images

G(x) =∑

m∈Z2

G(x + 2π(m, 0)), (22)

where m = (m1,m2) is a two-dimensional integer index. When the sum is writtenas in (22), it is strictly divergent. Since only the derivatives of the periodic Green’sfunction will be used, one way to alleviate this difficulty is to express this sum byusing the Ewald summation techniques as outlined by Baker, Meiron, and Orszag[2], which converts the derivatives of the periodic Green’s function into sums oferror functions. Alternatively, one can write this sum with a reflection and with aconstant subtracted from each term [4]

G(x) =12

∑m∈Z2,n6=0

(G(x + 2π(m, 0)) +G(x− 2π(m, 0)) +

12π|m|

). (23)

The gradient is

∇G(x) =12

∑m∈Z2

(∇G(x + 2π(m, 0)) +∇G(x− 2π(m, 0))) . (24)

It can be shown that both of these sums converge uniformly in L1 on bounded setsand G(x) gives the double periodic Green’s function [4].


3. An Unstable Boundary Integral Method for 3-D Water Waves. In thissection, we will demonstrate that the point vortex method approximation of the3D water wave equations (2)-(5) and (8)-(9) is numerically unstable. We denote byxj(t) the numerical approximation of x(αj , t), where αj = jh, j = (j1, j2) is a 2-Dinteger index, h is the mesh size. Similarly we define φj , µj , etc. We also need tointroduce a discrete derivative operator, Dh

l , which approximates ∂αl. This could

be a center difference derivative operator or a pseudo-spectral derivative operatorwith smoothing. For the convenience of our later analysis, we express the discretederivative operator in terms of the discrete Fourier transform. We recall that for a2π-periodic function,u, the discrete Fourier transform is given by

uk =h2

(2π)2

(N/2,N/2)∑(j1,j2)=(−N/2+1,−N/2+1)

u(αj)e−ik·xj , (25)

where h = 2π/N .The inversion formula is

uj =(N/2,N/2)∑

(k1,k2)=(−N/2+1,−N/2+1)

ukeik·xj . (26)

In terms of the discrete Fourier transform, we can express the discrete derivativeoperator as follows:

\(Dhl f)

k= iklρl(kh)fk, k1, k2 = −N

2+ 1, · · · , N

2, (27)

where fk is the discrete 2-D Fourier transform, and ρl(kh) is some non-negative cut-off function depending on the approximation being used. For example, ρl(kh) =sin(klh)

klhfor a second order center difference derivative operator. In the case of

spectral derivative, we require that ρl(x) = ρ(|x|) satisfies ρ(r) ≥ 0, ρ(π) = 0, andρ(r) = 1 for 0 ≤ r ≤ λπ with 0 < λ < 1. The Fourier smoothing is needed here toprevent the aliasing error in the stability analysis, as in the 2-D case [5]. We alsodenote by Sh

l the spectral derivative operator without smoothing, i.e.

\(Shl f)

k= iklfk, k1, k2 = −N

2+ 1, · · · , N

2. (28)

The point vortex method for 3D water waves is given as follows:

dxi

dt=

∑j6=i

ηj ×∇x′G(xi − xj)h2 +12

ηi × ni

|Dh1xi ×Dh

2xi|≡ wi, (29)

dφi

dt=

12|wi|2 − g · xi, (30)

µi = 2φi − 2∑j6=i

µjNj · ∇x′G(xi − xj)h2, (31)

where ηj = γ1jDh2xj − γ2jD

h1xj, Nj = Dh

1xj ×Dh2xj and nj = Nj

|Nj| , and γlj = Dhl µj

(l = 1, 2).To study the linear stability of the point vortex method, we first introduce the

following discrete convolution operators:


Hlh(fi) =12π

∑j6=i

(αli − αlj)fj((α1i − α1j)2 + (α2i − α2j)2)3/2

h2, l = 1, 2, (32)

and

Λh(fi) =12π

∑j6=i

(fi − fj)((α1i − α1j)2 + (α2i − α2j)2)3/2

h2. (33)

These operators will appear in the linear stability analysis around equilibrium.Moreover, we will also use the following discrete operator:

Rh = H1hDh1 +H2hD

h2 . (34)

We now examine the linear stability around equilibrium. Let x = (α1, α2, 0) +x′ and φ = 1

2 + φ′, where x′ and φ′ are assumed to be small. Substitute theseinto the point vortex method, (29)-(31), and neglect nonlinear terms. After somemanipulations, we obtain the equations that govern the evolution of the perturbedvariables as follows:

dx′idt

=(Dh

1µ′, Dh

2µ′, Rhµ

′) , (35)

dφ′idt

= −gz′i, (36)

where µ′ = φ′i + 12 (Λh −Rh)(z′i). We will consider the following equations

dz′idt

=(

0, 0, Rh(φ′i) +12Rh(Λh −Rh)(z′i)

), (37)

dφ′idt

= −gz′i, (38)

where z′ is the z-component of x′ = (x′, y′, z′). To study the growth rate of theabove linearized equations, we need to study the Fourier symbols of the discreteoperators, Hlh,Λh and Rh. Since these discrete operators are convolution operators,we can compute their Fourier symbols as follows:

(Hlh)k = − ikl

|k|bl(kh)fk, (39)

(Λh)k = |k|c(kh)fk, (40)

(Rh)k = |k|d(kh)fk, (41)(42)


where

b1(kh) =12π

∑j6=0

j1 sin(j1k1h) cos(j2k2h)(j21 + j22)3/2

, (43)

b2(kh) =12π

∑j6=0

j2 sin(j2k2h) cos(j1k1h)(j21 + j22)3/2

, (44)

c(kh) =1

2πh|k|∑j6=0

1− cos(k1j1h) cos(k2j2h)(j21 + j22)3/2

, (45)

d(kh) =k1b1ρ1 + k2b2ρ2

|k|. (46)

It can be shown that the above infinite series converge. Using the Fourier symbolsof the discrete operators, we can easily compute the eigenvalues of the linearizedsystem (37)-(38) as follows:

λ1, λ2 = 0, λ3, λ4 =C|k|2

4d(c− d)± 1

4

√|k|4d2(c− d)2 − 16g|k|d. (47)

If d(kh) 6= 0 and c(kh) 6= d(kh), then the unstable eigenvalues can grow as fastas O(|k|2) as |k| → ∞. Our numerical study has demonstrated convincingly that|d(kh)| and |c(kh)− d(kh)| are bounded away from a positive constant for the ma-jority of the Fourier modes. Thus the unstable eigenmodes can indeed grow expo-nentially fast in frequency space for t > 0 in a rate proportional to O(exp(c0|k|2t)for some positive constant c0. Numerical experiments for 3-D water waves nearequilibrium by David Haroldsen [13] also confirmed that numerical solutions of thepoint vortex method for small analytic perturbation from the equilibrium devel-oped order one oscillation in a short time even though the physical solution wasstill perfectly smooth at this time. The instability occurred earlier if a finer meshwas used.

It is clear that this instability is caused by violating the compatibility condition,Λ = H1D1 +H2D2, at the discrete level. While this compatibility can be imposedusing a Fourier filtering as in the 2-D case [5], there are four additional compatibilityconditions for 3-D water waves that need to be satisfied in order to obtain stabilityfar from equilibrium. We simply do not have enough degree of freedom in filteringthe interface variables to satisfy all these compatibility conditions. This is why weneed to introduce a different stabilizing technique to obtain a stable 3-D boundaryintegral method.

4. A New Stabilizing Technique for 3-D Water Waves. As we mentionedin the introduction, it is important to separate the treatment of linear stabilityfrom that of nonlinear stability when we consider stability of a numerical methodfor a well-posed initial value problem. It is linear stability that plays the mostimportant role in obtaining convergence of a boundary integral method, as longas the solution is sufficiently smooth. Our stability analysis has two importantingredients. First we derive a leading order approximation of the singular velocityintegral. This leading order approximation captures all the leading order contri-butions of the original velocity integral to linear stability analysis. Moreover, theleading order approximation can be expressed in terms of the Riesz transform, andcan be approximated with spectral accuracy. Thus the spectral properties of theoriginal velocity integral are preserved exactly at the discrete level by this leading


order approximation. Secondly, the term consisting of the difference of the originalvelocity integral and the leading order approximation is less singular and does notcontribute to the linear stability. Thus it can be approximated by any consistentquadrature rule such as the point vortex method approximation. Since the leadingorder approximation in effect has desingularized the velocity integral, we obtain animproved 3rd order accuracy of the (modified) point vortex method approximationfor 3-D water waves. The stabilizing technique described above can also be inter-preted as a near field correction to the point vortex approximation. Although thenear field correction is small in ampltitude, O(h), it contains critical high frequencycontribution which stabilizes the point vortex approximation.

The leading order approximation of the velocity integral needs to satisfy twoproperties: (i) It captures all the leading order contributions of the original velocityintegral, (ii) It can be evaluated with spectral accuracy. The nonlocal leading orderapproximations for xαl

·w0, l = 1, 2 and w0 · n which satisfy these two criteria aregiven by

Bl =N(α)

2· (γ1(α)H1 + γ2(α)H2)xαl

(α), l = 1, 2 (48)

and

Bn =|N(α)|

2{[H1(γ1) +H2(γ2)]

−γ1(α)x∗α1· (H1D1 +H2D2)x(α)

−γ2(α)x∗α2· (H1D1 +H2D2)x(α)} , (49)

where H1 and H2 are the Riesz transforms defined in (67) in section 6. The Fouriersymbols of the Riesz transforms can be computed explicitly (see section 6). Thusthese singular integrals can be evaluated via discrete Fourier transform with spectralaccuracy. Moreover, using a special coordinate frame with the property [16]

xα1 · xα2 = λ1(t)|xα2 |2, |xα1 |2 = λ2(t)|xα2 |2, (50)

the Riesz transforms become convolution operators and can be evaluated by FastFourier Transform with O(N2 log(N)) operation counts.

In addition to using a near field correction described above, we also use desin-gularization to stabilize the aliasing error introduced by the point vortex approxi-mation. In our desingularization, we use the following two identities [2]∫

N(α′) · ∇x′G(x(α)− x(α′))dα′ = 0, (51)∫∇x′G(x(α)− x(α′))×N(α′)dα′ = 0. (52)

The use of these identities to reduce the singularity was suggested in [2] and [4].We define

Cl = γl(α)∫

N(α′) · ∇x′G(x(α)− x(α′))dα′, l = 1, 2 (53)

Cn =η(α)|N(α)|

·∫∇x′G(x(α)− x(α′))×N(α′)dα′. (54)

It follows from (51) and (52) that Cl = 0, l = 1, 2 and Cn = 0.


Now we are ready to describe our discretization for the 3-D water wave equations.First we describe the approximation of the normal velocity integral w0 · n:

wh0 · nh − Ch

n + (Bsn −Bh

n) , (55)

where wh0 , C

hn , B

hn are the point vortex approximations of w0, Cn, Bn respectively,

Bsn is the spectral approximation of Bn, and

wh0 =

∑j6=i

ηj ×∇x′G(xi − xj)h2,

Chn =

ηi|Ni|

·∑j6=i

Nj ×∇x′G(xi − xj)h2,

where ηj = γ1jDh2xj − γ2jD

h1xj, Nj = Dh

1xj ×Dh2xj.

In the discretization of (55), the second term, −Chn , is a desingularizing term

and is introduced to remove the aliasing error associated with the point vortexapproximation of w0 · n. As noted by Beale in [4], the desingularizing term helpsprevent spurious terms from appearing in the stability analysis. The last term in(55) is a near field correction term. This term is small (of order O(h)), but itcontributes to the stability of the discretization in an essential way.

Next we describe the approximation of the tangential velocity integral xαl·

w0, l = 1, 2:

Dhl xi ·wh

0 − Chl + (Bs

l −Bhl ), l = 1, 2 (56)

where wh0 , C

hl , B

hl are the point vortex approximations of w0, Cl, Bl respectively,

Bsl is the spectral approximation of Bl, and

Chl = γli

∑j 6=i

Nj · ∇x′G(xi − xj)h2, l = 1, 2.

As in the discretization of the normal velocity, the second term in (56), −Chl , is a

desingularizing term and is introduced to remove the aliasing error associated withthe point vortex approximation of Dh

l xi ·w0. The last term in (56) is a near fieldcorrection term, which contributes to the stability of the discretization.

Now we can state our modified point vortex method for the 3-D water waveproblem.

dxi

dt= Dh

1φiDh1x∗i +Dh

2φiDh2x∗i + (wh

0 · nhi − Ch

n +Bsn −Bh

n)ni ≡ wi, (57)

dφi

dt=

12|wi|2 − g · xi, (58)

Dhl φi =

γli

2+ Ph

l1(Dh1xi ·wh

0 − Ch1 + (Bs

1 −Bh1 )) + Ph

l2(Dh2xi ·wh

0 − Ch2

+(Bs2 −Bh

2 )), l = 1, 2 (59)

where Ni = Dh1xi × Dh

2xi and ni = Ni

|Ni| , Dhl is a spectral derivative operator,

and Phlk, l, k = 1, 2 is a spectral approximation of Plk, l, k = 1, 2, which is defined

through the discrete Fourier transform of Phlk as follows: (Ph

ij)k = kikj

|k|2 for k 6= 0,

where k = (k1, k2), and (Phij)k = 0 for k = 0.

Note that because we project the tangential velocity field into x∗α1and x∗α2

, weobtain a simplified expression for the discretization of the interface equation (57).On the other hand, since we use the reformulation for the γ1 and γ2 equations which


involve the tangential velocity xαi·w0, we need to use the near field correction and

the desingularization to the tangential velocity field to obtain stability.The main result of this paper is the following convergence result.

Theorem 1. Assume that the water wave problem is well-posed and has asmooth solution in CM (M ≥ 6) up to time T . Then the modified point vortexmethod (57)-(59) is stable and convergent. More precisely, there exists a positiveh0(T ) such that for 0 < h ≤ h0(T ) we have

‖x(t) − x(·, t)‖l2 ≤ C(T )h3 , (60)‖γl(t) − γl(·, t)‖l2 ≤ C(T )h3, l = 1, 2. (61)

Here ‖x‖2l2 =∑N

i,j=1 |xi,j |2h2 .

The proof of Theorem 1 will be deferred to Section 5.

5. Consistency of the modified point vortex method. In this section, wewill prove the consistency of the modified point vortex method for 3-D water waves.Since we approximate the derivative operator and the leading order approximationof the velocity integral with spectral accuracy, it is sufficient to prove the consistencyof the point vortex method approximation of the desingularized tangential andnormal velocity integrals xαl

· w0, l = 1, 2 and w0 · n . We will show that themodified point vortex method approximation is third order accurate and has anerror expansion in the odd powers of h. First, we introduce a function vl(α, α′), l =1, 2 and vn(α, α′) as follows:

vl(α, α′) = xαl(α) · η(α′)×∇x′G(x(α)− x(α′))

−γl(α)N(α′) · ∇x′G(x(α)− x(α′))

−N(α)2

· (γ1(α)α1 − α′1r3

+ γ2(α)α2 − α′2r3

)xαl(α), l = 1, 2,(62)

and

vn(α, α′) = n(α) · η(α′)×∇x′G(x(α)− x(α′))

− η(α)|N(α)|

· ∇x′G(x(α)− x(α′))×N(α′)

−|N(α)|2

((α1 − α′1r3

γ1 +α2 − α′2r3

γ2)(α)

−γ1(α)2

x∗α1(α) · (α1 − α′1

r3D1 +

α2 − α′2r3

D2)x(α)

−γ2(α)2

x∗α2(α) · (α1 − α′1

r3D1 +

α2 − α′2r3

D2)x(α)) , (63)

where r = |r|, r = xα1(α)(α1−α′1)+xα2(α)(α2−α′2). It is easy to see that vl(α, α′)and vn(α, α′) are the corresponding continuous integrands in the desingularizedpoint vortex approximations of the tangential and normal components of the in-terface velocity. Then the consistency of the point vortex method approximationof the desingularized tangential and normal velocity integrals is reduced to provingthat∫ ∫

vl(αi, α′)dα′ −

∑j6=i

vl(αi, αj)h2 = C3h3 + C5h

5 + · · ·+ C2n+1h2n+1 + · · · .(64)


for l = 1, 2, and similarly for the normal component. The error expansion (64)can be proved by using an argument similar to that in [12]. We first break thiserror estimate into far field and near field estimates using a smooth cut-off function,fδ(|α|), which satisfies (i) fδ(|α|) = 1 for |α| ≤ δ/2, (ii) fδ(|α|) = 0 for |α| ≥ δ.We will take δ to be small, but independent of h. For the far field estimate, whichamounts to replacing vl by vl(α, α′)gδ(|α − α′|) with gδ = 1 − fδ, classical erroranalysis shows that the far field error is of spectral accuracy, O(hM ) (M is thedegree of regularity of x and γ), see, e.g. [9]. For the near field error, we can Taylorexpand vl(α, α′) around α′ = α. Without loss of generality, we may assume thatαi = 0. It is not difficult to show that for |α′| ≤ δ small

vl(0, α′) = m−2(α′) + m−1(α′) + m0(α′) + m1(α′) + · · · (65)

where ml(α) (l = −2,−1, 0, 1, · · · ) are homogeneous functions of degree l, andml(α) is odd function of α for l even. Since the cut-off function fδ(|α|) is an evenfunction of α, ml(α)fδ(|α|) is also an odd function of α for l even.

To justify the expansion (65), we Taylor expand γl(α′), xαl(α′), and∇x′G(x(0)−

x(α′)) around α′ = 0. We have

γl(α′) = γl(0) + (γl)α(0)(α′) + m2(α′) + · · · ,xαl

(α′) = xαl(0) + (xαl

)α(0)(α′) + m2(α′) + · · · ,

∇x′G(x(0)− x(α′)) = ∇x′G(r(0))−∇x′∇x′G(r(0))(12(α′)T∇∇x(0)(α′))

+m0(α′) + · · · ,where r(α) = D1x(α)(α1−α′1) +D2x(α)(α2−α′2) and ml(α) is a generic notationfor a homogeneous function of degree l. Substituting the above expansions intothe (62), and collecting the terms of the same order give rise to the expansion(65). Moreover, direct calculations show that the homogeneous terms of degree−1 exactly cancel each other. This implies that m−1(α) ≡ 0 in the expansion(65). Since both m−2 and m0 are odd, they do not contribute to the continuousintegral nor the discrete sum. Therefore the first term in the expansion of vl thatcontributes to the error is the m1(α) term.

We now show that∫ml(α′)fδ(|α′|)dα′ −

∑j6=0

ml(αj)fδ(|αj|)h2 = Cl+2hl+2 +O(hM ), for l ≥ 1. (66)

The proof follows closely that of [12]. Using the homogeneity of ml, we can write∑j6=0

ml(αj)fδ(|αj|)h2 = hl+2Sl(h), with Sl(h) =∑j6=0

ml(j)fδ(|j|h).

Since Sl is a finite sum due to the cut-off fδ, we may differentiate Sl(h) with respectto h. We have

d

dhSl(h) =

∑j6=0

ml(j)d

drfδ(|j|h)|j| = h−(l+3)

∑j6=0

ml(jh)d

drfδ(|j|h)|jh|h2.

The sum is a trapezoidal rule approximation to∫ml(y)

d

drfδ(|y|)|y|dy.

Note that the integrand is a smooth function with compact support since ddrfδ(|y|)

vanishes near the origin and has a compact support. It is well-known that the


trapezoidal rule gives spectral accuracy for such smooth integrand with compactsupport [9]. That is∑

j6=0

ml(jh)d

drfδ(|j|h)|jh|h2 =

∫ml(y)

d

drfδ(|y|)|y|dy +O(hM ).

On the other hand, using the polar coordinate, we obtain∫ml(y)

d

drfδ(|y|)|y|dy =

∫ 2π

0

∫ ∞

0

rlml(θ)d

drfδ(r)r2drdθ

=∫ 2π

0

ml(θ)dθ∫ ∞

0

rl+2 d

drfδ(r)dr

= −(l + 2)∫ 2π

0

ml(θ)dθ∫ ∞

0

rlfδ(r)rdr

= −(l + 2)∫

ml(y)fδ(|y|)dy,

where we have used the homogeneity of ml. Thus we haved

dhSl(h) = −(l + 2)h−(l+3)

∫ml(y)fδ(|y|)dy +O(hM ).

Integrating the above equation from h to 1 gives

Sl(h) = −Cl+2 + h−(l+2)

∫ml(y)fδ(|y|)dy +O(hM ),

where Cl+2 is an integration constant. This implies that∫ml(y)fδ(|y|)dy −

∑j6=0

ml(αj)fδ(|αj|)h2 =∫

ml(y)fδ(|y|)dy − h(l+2)Sl(h)

= Cl+2h(l+2) +O(hM ),

for all l ≥ 1. Observe that Cl = 0 for l even due to the oddness of ml. Thus weobtain an error expansion in the odd powers of h. This proves the consistency ofthe modified point vortex method.

6. Properties of some singular integral operators. Before we proceed to an-alyze the stability of our 3-D boundary integral method, we need to study thespectral properties of certain singular integral operators. We first define the Riesztransform (which is the 2-D analogue of the Hilbert transform) and the Λ operator

Hl(f) =12π

∫ ∫(αl − α′l)f(α′)

r3dα′, l = 1, 2, (67)

Λ(f) =12π

∫ ∫f(α)− f(α′)

r3dα′, (68)

where r = |r|, and

r = xα1(α)(α1 − α′1) + xα2(α)(α2 − α′2).

The Fourier transform of a 2-D function f is defined by

f(ξ) = (2π)−2

∫ ∫f(x)e−ix·ξdx. (69)


The inverse transform is given by

f(x) =∫ ∫

f(ξ)eix·ξdξ. (70)

Note that the Riesz transforms defined in (67) are not a convolution operator.They are pseudo-differential operators. Their ‘Fourier symbols’ depend on the freesurface x(α). Nonetheless, we can still characterize their spectral properties via theFourier transform. The result is summarized in the following lemma.

Lemma 6.1. The Riesz transforms have the following spectral representation:

(H1f)(α) =−i

|xα1 × xα2 |2

∫ ∫(|xα2 |2ξ1 − (xα1 · xα2)ξ2)f(ξ)eiξ·α

(|xα2 |2ξ21 − 2(xα1 · xα2)ξ1ξ2 + |xα1 |2ξ22)1/2dξ,

(71)

(H2f)(α) =−i

|xα1 × xα2 |2

∫ ∫(|xα1 |2ξ2 − (xα1 · xα2)ξ1)f(ξ)eiξ·α

(|xα2 |2ξ21 − 2(xα1 · xα2)ξ1ξ2 + |xα1 |2ξ22)1/2dξ.

(72)

Proof: In the special case of an orthogonal parametrization of the free surface, i.e.xα1 · xα2 = 0, this result has been obtained in Lemma 3.1 of [18]. The result fora non-orthogonal parametrization can be obtained in a similar way. We express fin terms of his Fourier transform and substitute it into the Riesz transforms. Weobtain

(H1f)(α) =12π

∫ ∫f(ξ)dξ

∫ ∫(α1 − α′1)e

iξ·α′

r3dα′

=12π

∫ ∫f(ξ)eiξ·αdξ

∫ ∫α′1e

−iξ·α′

(|xα1 |2α′21 + 2xα1 · xα2α

′1α

′2 + |xα2 |2α

′22 )3/2

dα′.

(73)

We define

Φ(ξ1, ξ2) =∫ ∫

α′1e−iξ·α′

(|xα1 |2α′21 + 2xα1 · xα2α

′1α

′2 + |xα2 |2α

′22 )3/2

dα′.

By making a change of variables from α′ to β′,

β′1 =|xα1 × xα2 ||xα2 |

α′1, β′2 =xα1 · xα2

|xα2 |α′1 + |xα2 |α′2,

and using the identity

|xα1 × xα2 |2 + (xα1 · xα1)2 = |xα1 |2|xα2 |2, (74)

we can show that

Φ(|xα1 × xα2 ||xα2 |

ξ1 +xα1 · xα2

|xα2 |ξ2, |xα2 |ξ2) =

|xα2 ||xα1 × xα2 |2

∫ ∫β′1e

−iξ·β′

(β′21 + β

′22 )3/2

dβ′.

It has been shown in [23] (pp. 57-58) that the Fourier transform of the Riesz kernelβ1

(β21+β2

2)3/2 is given by∫ ∫β′1e

−iξ·β′

(β′21 + β

′22 )3/2

dβ′ = −2πiξ1

(ξ21 + ξ22)1/2.


Therefore, we obtain

Φ(ξ1, ξ2) =−2πi

|xα1 × xα2 |2(|xα2 |2ξ1 − (xα1 · xα2)ξ2)

(|xα2 |2ξ21 − 2(xα1 · xα2)ξ1ξ2 + |xα1 |2ξ22)1/2.

By (73), we have

(H1f)(α) =−i

|xα1 × xα2 |2

∫ ∫(|xα2 |2ξ1 − (xα1 · xα2)ξ2)f(ξ)eiξ·α

(|xα2 |2ξ21 − 2(xα1 · xα2)ξ1ξ2 + |xα1 |2ξ22)1/2dξ.

Using the same argument, we can get the expression of (H2f)(α). This provesLemma 6.1.

Lemma 6.2. The Λ operator satisfies the following compatibility condition:

Λ = H1D1 +H2D2. (75)

where Dl = ∂αl. Moreover, ΛHl is a local derivative operator to the leading order:

ΛH1(f) = − 1|xα1 × xα2 |4

(|xα2 |2D1 − (xα1 · xα1)D2

)f +A0(f), (76)

ΛH2(f) = − 1|xα1 × xα2 |4

(|xα1 |2D2 − (xα1 · xα2)D1

)f +A0(f), (77)

where A0(f) is a bounded operator from Hs to Hs.

Proof: The identity (75) can be verified directly by using integration by parts.Using (75) and Lemma 6.1, we obtain

(Λf)(α) =1

|xα1 × xα2 |2

∫(|xα2 |2ξ21 − 2(xα1 · xα2)ξ1ξ2 + |xα1 |2ξ22)1/2f(ξ)eiξ·αdξ. (78)

Define the commutator operator as follows:

[Λ, g](f) = Λ(gf)− gΛ(f).

It is easy to show that the commutator operator [Λ, g] = A0 is a bounded operatorfor g smooth. Now using (78), Lemma 6.1, and [Λ, g] = A0, we have

ΛH1(f) = − i

|xα1 × xα2 |4

∫(|xα2 |2ξ1 − (xα1 · xα2)ξ2)f(ξ)eiξ·αdξ +A0(f)

= − |xα2 |2

|xα1 × xα2 |4

∫ ∫iξ1f(ξ)eiξ·αdξ +

xα1 · xα2

|xα1 × xα2 |4

∫ ∫iξ2f(ξ)eiξ·αdξ

+A0(f)

= − 1|xα1 × xα2 |4

(|xα2 |2D1 − (xα1 · xα1)D2

)f +A0(f).

This proves (76). The proof of (77) follows similarly. This completes the proof ofLemma 6.2.

Now we can describe the spectral discretization of Hl, l = 1, 2 defined in section4. Using the explicit Fourier representation of Hl in Lemma 6.1, we can defineour spectral discretization of the Hl operator, denoted as Hs

l . To this end, it issufficient to give a spectral discretization of the Riesz transform. We denote by Hs

l

the spectral discretization of the Riesz transform Hl,

(Hs1,hf)(αj) = − i

|Dh1x×Dh

2x|2∑

k∈KN

(|Dh2x|2k1 − (Dh

1x ·Dh2x)k2)fkeik·αj

(|Dh2x|2k2

1 − 2(Dh1x ·Dh

2x)k1k2 + |Dh1x|2k2

2)1/2,


(Hs2,hf)(αj) = − i

|Dh1x×Dh

2x|2∑

k∈KN

(|Dh1x|2k2 − (Dh

1x ·Dh2x)k1)fkeik·αj

(|Dh2x|2k2

1 − 2(Dh1x ·Dh

2x)k1k2 + |Dh1x|2k2

2)1/2,

where Dhl x = Dh

l x(αj) and KN = {k : k 6= 0,−N/2 + 1 ≤ kl ≤ N/2, l = 1, 2}.

Remark 6.1 If we evaluate directly the above spectral discretization of the Riesztransform, it will cost O(N4 logN) operations which are comparable to the oper-ation count of the point vortex method approximation using direct summations.On the other hand, it is possible to reduce the Riesz transform, Hl, to a convolu-tion operator by introducing a generalized arclength parametrization for the freesurface. Such a generalized arclength parametrization can be obtained by imposing

xα1 · xα2 = λ1(t)|xα2 |2, |xα1 |2 = λ2(t)|xα2 |2.

It can be shown that such a parametrization exists, at least for near equilibriuminitial data [16]. Our numerical experiments have indicated that this parametriza-tion exists quite generically even for large perturbations from the equilibrium [16].Once we have found such parametrization of the free surface initially, this propertyis preserved in time by adding two tangential velocities dynamically. We refer thereader to [16] for more details. Using this generalized arclength frame, the Riesztransform now becomes a convolution operator. In fact, we now have

(Hs1,hf)(αj) = − |Dh

2x| i|Dh

1x×Dh2x|2

∑k∈KN

(k1 − λ1(t)k2)fkeik·αj

(k21 − 2λ1(t)k1k2 + λ2(t)k2

2)1/2,

(Hs2,hf)(αj) = − |Dh

2x| i|Dh

1x×Dh2x|2

∑k∈KN

(k2 − λ1(t)k1)fkeik·αj

(k21 − 2λ1(t)k1k2 + λ2(t)k2

2)1/2,

which can be evaluated by Fast Fourier Transform with O(N2 logN) operationcount.

7. Stability of the modified point vortex method for 3-D water waves. Inthis section, we will analyze stability the modified point vortex method. We firstconsider the linear variation of the modified point vortex method approximation tothe tangential and normal velocity integral, i.e. the terms xαl

·w0, l = 1, 2 and w0·n.Denote the errors in xj, γlj and φj by xj = xj−x(αj), γlj = γlj−γl(αj), l = 1, 2, andφj = φj − φ(αj) respectively. Epvm(j), Enear(j), and Elocal(j) are defined similarly.We sometimes also use the notation, Er(f)(j) = f(αj) − fj when the quantity fconsists of several terms. Further, we define

El(αi) ≡(−xαl

(α) ·∫ ∫

η(α′)×∇x′G(x(α)− x(α′))dα′

+Dhl xi ·

∑j6=i

ηj ×∇x′G(xi − xj)h2

−γli

∑j6=i

Nj · ∇x′G(xi − xj)h2

+ (Bsl −Bp

l )

≡ Elpvm + El

near, l = 1, 2, (79)


and

En(αi) ≡(−n(α) ·

∫ ∫η(α′)×∇x′G(x(α)− x(α′))dα′

+ni ·∑j6=i

ηj ×∇x′G(xi − xj)h2

− ηi

|Ni|·∑j6=i

∇x′G(xi − xj)×Njh2

+ (Bsn −Bp

n)

≡ Enpvm + En

near, (80)

where Bpl , l = 1, 2,n denotes the point vortex method approximation of Bl, and

Bsl denotes the spectral approximation of Bl using the exact Fourier symbols of

Bl. Stability analysis of our approximations to the tangential and normal velocityintegrals is to estimate El and En in terms of xj and γlj, etc.

The first term on the right hand side of (79) and (80) measures the stabilityerror due to the point vortex method approximation of the velocity integral. So wecall it El

pvm (l = 1, 2) and En. The second term on the right hand side is due tothe contribution of a near field correction. So we term it El

near (l = 1, 2) and En.We first study the linear variation of El

pvm, l = 1, 2. Direct calculations showthat the linear variation of El

pvm, l = 1, 2, denoted as Elpvm, is given by

E1pvm(i) = Dh

1 xi ·wh0 +Kh

1 γ1 −Kh2 γ2

+Ni

2· (γ1iH

p1,h + γ2iH

p2,h)Dh

1 xi

+O(h)(γi) +A0(xi), (81)

E2pvm(i) = Dh

2 xi ·wh0 +Kh

3 γ1 −Kh4 γ2

+Ni

2· (γ1iH

p1,h + γ2iH

p2,h)Dh

2 xi

+O(h)(γi) +A0(xi), (82)

Enpvm(i) = n ·wh

0 +|Ni|2

(Hp

1,h(γ1i) +Hp2,h(γ2i)

−γ1iDh1x∗i · (H

p1,hD

h1 +Hp

2,hDh2 )xi

−γ2iDh2x

∗i · (H

p1,hD

h1 +Hp

2,hDh2 )xi

)+A−1(γi) +A0(xi), (83)

where γli = γl(αi) (l=1,2), Khl is a point vortex method approximation of Kl

(l = 1, 2, 3, 4), wh0 is a point vortex method approximation of w0, A0 is a bounded

operator from lp to lp, and A−1 is a smoothing operator of order one, i.e. Dhl A−1 =

A−1(Dhl ) = A0. We defer to Appendix C to give a more detailed derivation of the

above error estimates.Next we estimate the linear variation of the near field term Enear. Direct calcu-

lations show that the linear variation of Enear, denoted as Elnear(l=1,2) and En

near,


is given by

E1near(i) =

Ni

2· (γ1iH

s1,h + γ2iH

s2,h)Dh

1 xi

−Ni

2· (γ1iH

p1,h + γ2iH

p2,h)Dh

1 xi

+O(h)(γi) +A0(xi). (84)

E2near(i) =

Ni

2· (γ1iH

s1,h + γ2iH

s2,h)Dh

2 xi

−Ni

2· (γ1iH

p1,h + γ2iH

p2,h)Dh

2 xi

+O(h)(γi) +A0(xi). (85)

Ennear(i) =

|Ni|2

(Hs

1,h(γ1i) +Hs2,h(γ2i)

−γ1iDh1x∗i · (Hs

1,hDh1 +Hs

2,hDh2 )xi


1,hDh1 +Hs

2,hDh2 )xi

)−|Ni|

2

(Hp

1,h(γ1i) +Hp2,h(γ2i)


p1,hD

h1 +Hp

2,hDh2 )xi


p1,hD

h1 +Hp

2,hDh2 )xi

)+A−1(γi) +A0(xi) . (86)

Again, we defer to Appendix C to give a more detailed derivation of the above errorestimates.

By adding Elpvm and El

near (l = 1, 2) and adding Enpvm and En

near, we obtainafter some cancellations that

E1pvm(i) + E1

near(i) = Dh1 xi ·wh

0 +Kh1 γ1 −Kh

2 γ2

+Ni

2· (γ1iH

s1,h + γ2iH

s2,h)Dh

1 xi

+O(h)(γi) +A0(xi), (87)

E2pvm(i) + E2

near(i) = Dh2 xi ·wh

0 +Kh3 γ1 −Kh

4 γ2

+Ni

2· (γ1iH

s1,h + γ2iH

s2,h)Dh

2 xi

+O(h)(γi) +A0(xi), (88)

Enpvm(i) + En

near(i) = n ·wh0 +

|Ni|2

(Hs

1,h(γ1i) +Hs2,h(γ2i)


1,hDh1 +Hs

2,hDh2 )xi


1,hDh1 +Hs

2,hDh2 )xi

)+A−1(γi) +A0(xi). (89)

Projection of errors into local normal and tangent vectors.

As we can see from (87)–(89), the leading order error terms of the tangential andnormal velocity components can be naturally expressed in terms of the local normaland tangent vectors. This suggests that we project the interface errors into the localnormal and tangent vectors instead of using the original (x, y, z) coordinate. In the


local normal and tangent coordinates, the equations that govern the growth of theleading order errors greatly simplify and we can identify cancellation of interfaceerrors in certain tangent direction. This makes it easier to obtain stability byperforming energy estimates.

On the other hand, we have from (59) that

Dhl φi =

γli

2+ Ph

l1Er((Dh1xi ·wh

0 − Ch1 ) + (Bs

1 −Bh1 ))

+ Phl2Er((D

h2xi ·wh

0 − Ch2 ) + (Bs

2 −Bh2 )), (90)

where Er(f) = f(αj) − fj denotes the error of f . Using (87) and (88) , and thedefinition of Epvm(i) and Enear(i), we get(

I

2+Ah +O(h)

)γi = 5hφi −5hxi ·wh

0 +Ni

2· (γ1iH

s1,h + γ1iH

s1,h)5h xi (91)

where Ah is two by two matrix, Ah = (ahij) with aij given by

ah11 = Ph

11Kh1 + Ph

12Kh3 , ah

12 = Ph11K

h2 + Ph

12Kh4 ,

ah21 = Ph

21Kh1 + Ph

22Kh3 , ah

22 = Ph21K

h2 + Ph

22Kh4 .

We can show that 12I + Ah + O(h) is invertible by using the invertibility of the

corresponding continuous operator and the fact that the kernel of Ah approximatesthat of A to the leading order. The following lemma, which is proved in AppendixA, concerns the solvability of (91).

Lemma 7.1. Given x(α, t) smooth for 0 ≤ t ≤ T , the operator I+2Ah is invertiblefor h sufficiently small, and the norm of (I+2Ah)−1 as an operator on L2

h is boundeduniformly in h and t.

Since Kl is an integral operator with weakly singular kernel, one can verify thatKh

l (φ) is a smoothing operator of order one, i.e. Khl (φ) = A−1(φ) (l = 1, 2, 3, 4).

We defer to Appendix B to prove this property (see also [4] for proof of a similarresult). We can show that the invertibility of (1

2I + Ah) and the fact Khl (φ) =

A−1(φ) (l = 1, 2, 3, 4) imply

γli

2= Dh

l φi −(Dh

l xi ·w0(αi) +Ni

2· (γ1iH

s1,h + γ2iH

s2,h)Dh

l xi

)+ A0(φi) +A0(xi), (92)

which can be rewritten as

γli

2= Dh

l (Fi + Elocal(αi) · xi) +Ni

2· (γ1iH

s1,h + γ2iH

s2,h)Dh

l xi

+A0(xi) +A0(Fi), (93)

where

Fi = φi −w(αi) · xi, (94)


and

Elocal(α) = w(α)−w0(α)= φα1x

∗α1

+ φα1x∗α2

+ (w0 · n)n−((w0 · xα1)x

∗α1

+ (w0 · xα2)x∗α2

+ (w0 · n)n)= ((φα1 −w0 · xα1)x

∗α1

+ (φα2 −w0 · xα2)x∗α2

=γ1

2x∗α1

+γ2

2x∗α2

. (95)

It follows from (93) that A−1(γi) = A0(Fi) +A0(xi). From the definition of Elocal,we have

Elocal · xi =12

(γ1iD

h1x∗i + γ2iD

h2x∗i

)· xi. (96)

Substituting (93) into (89) and using (96), we obtain after some cancellations that

Enpvm(i) + En

near(i) = n ·wh0 + |Dh

1xi ×Dh2xi|(Hs

1,hDh1 +Hs

2,hDh2 )Fi

+|Ni|2

2ni · (γ1iH

s1,h + γ2iH

s2,h)(Hs

1,hDh1 +Hs

2,hDh2 )xi

+ A0(xi) +A0(Fi). (97)

Furthermore, direct calculations show that

Elocal(i) · ni =γ1i

2|Ni|2((Dh

2x× n)i ×Dh2xi ·Dh

1 xi − (Dh2x× n)i ×Dh

1xi ·Dh2 xi

)+

γ2i

2|Ni|2((n×Dh

1x)i ×Dh1xi ·Dh

2 xi − (n×Dh1x)i ×Dh

2xi ·Dh1 xi

)=

γ1i

2|Ni|2ni ·

(|Dh

2xi|2Dh1 xi − (Dh

2xi ·Dh1xi) ·Dh

2 xi

)+

γ2i

2Ni|2ni ·

(|Dh

1xi|2Dh2 xi − (Dh

1xi ·Dh2xi)Dh

1 xi

). (98)

Using (97), (98) and Lemma 4.2, we obtain after canceling the local terms that

wi · ni = Er(Dh

1φiDh1x

∗i +Dh

2φiDh2x∗i + (wh

0 · nhi − Ch

n +Bsn −Bh

n)ni

)· ni

= Er(Dh1φiD

h1x∗i +Dh

2φiDh2x∗i ) · ni + En

pvm(i) + Ennear(i)

= −(Dh1φiD

h1x∗i +Dh

2φiDh2x∗i ) · ni + En

pvm(i) + Ennear(i)

= −Elocal · ni −(wh

0 ·Dh1xi)Dh

1x∗i + (wh0 ·Dh

2xi)Dh2x∗i

)· ni + En

pvm(i)

+Ennear(i)

= Elocal · ni −wh0 · ni + En

pvm(i) + Ennear(i)

= |Dh1xi ×Dh

2xi|(Hs1,hD

h1 +Hs

2,hDh2 )Fi +A0(x) +A0(F ). (99)

Here Er(f) = f(αj)−fj as it was defined earlier. This completes our error estimatefor the normal velocity component.

The error estimate for the tangential velocity component is much easier. Notethat equation (57) implies

Dhl φi = Dh

l xi ·wi.

Thus we have

Dhl φi = Dh

l xi ·wi +Dhl xi · wi, (100)


which implies

wi ·Dhl x(αi) = Dh

l (F ) +A0(xi). (101)

This completes the error estimate for the tangential velocity component.

Derivation of the error equations in the normal and tangent coordinates.

We are ready to perform energy estimates to obtain stability. As in the 2-D case [5], we need to first derive the error equations in the local normal andtangent coordinates. Let t1, t2, and n be the local unit tangent and normal vectorsrespectively. Denote by xtl

= x · tl (l = 1, 2), and xn = x · n. After projecting theerror equations into the local tangent and normal vectors, we obtain

∂xt1

i

∂t=

1|Dh

1xi|Dh

1 Fi +A0(x) (102)

∂xt2

i

∂t=

1|Dh

2xi|Dh

2 Fi +A0(x) (103)

∂xni

∂t= |Dh

1xi ×Dh2xi|

(Hs

1,hDh1 +Hs

2,hDh2

)Fi +A0(x) +A0(F ), (104)

where we have used (99)-(101).we will need to express the evolution equation for Fi in the new variables. From

the definition of Fi (see (94)), we have

(Fi)t = (φi)t −w(αi) · (xi)t −wt(αi) · xi. (105)

To evaluate (φi)t, we compare the continuous Bernoulli equation with its discreteapproximation, i.e.

(φi)t =12|wi|2 − g · xi, (106)

(φ(αi))t =12|w(αi)|2 − g · x(αi). (107)

Subtracting (107) from (106), we obtain

(φi)t = wi ·w(αi)− g · xi +12|wi|2. (108)

Substituting (108) into (105), we have after some cancellations

(Fi)t = −(wt(αi) + g) · xi +12|wi|2. (109)

Noting that the Lagrangian velocity w satisfies the Euler equations in the fluiddomain in the form

wt = −∇p− g.Thus we have

(Fi)t = ∇p · xi +12|wi|2. (110)

Moreover, p = 0 on the interface so that ∇p is in the normal direction. We have

∇p · xi = −c(αi)xni , (111)

where c(α) = −∇p · n = (wt + g) · n. Hence the evolution equation for F becomes

(Fi)t = −c(αi)xni +

12|wi|2. (112)


Now the whole set of evolution equations for xtl

(l = 1, 2), xn, and F is given by

∂xt1

i

∂t=

1|Dh

1xi|Dh

1 Fi +A0(x) (113)

∂xt2

i

∂t=

1|Dh

2xi|Dh

2 Fi +A0(x) (114)

∂xni

∂t= |Dh

1xi ×Dh2xi|

(Hs

1,hDh1 +Hs

2,hDh2

)Fi +A0(x) +A0(F ) (115)


12|wi|2. (116)

Thus, we have reduced the error estimates for the full nonlinear, nonlocal water waveequations into a simple linear and almost local system for the variation quantities.The only nonlocality in the leading order terms comes from the discrete Riesztransform Hs

l,h(l = 1, 2). The lower order terms are nonlocal, but they are smootherthan the principal linearized terms. This simplification helps us identify and balancethe most important terms in our energy estimates.

The energy estimates (113)-(116) can be obtained directly. As in the two dimen-sional case, the normal variation plays an essential role. The tangential variationsonly contribute to the lower order terms. To see this, we make the following changeof variables for the tangential variations xt1

i and xt2

i :

δ1i = xt1

i +|Dh

1xi ×Dh2xi|

|Dh1xi|

[|Dh

1xi|2Hs1,h + (Dh

1xi ·Dh2xi)Hs

2,h

]xni (117)

δ2i = xt2

i +|Dh

1xi ×Dh2xi|

|Dh2xi|

[|Dh

2xi|2Hs2,h + (Dh

1xi ·Dh2xi)Hs

1,h

]xni . (118)

Using Lemma 4.2 and the identity (74), we have

Λ(|xα1 |2H1 + (xα1 · xα2)H2)(f) = − 1|xα1 × xα1 |2

D1f +A0(f) (119)

Λ(|xα2 |2H2 + (xα1 · xα2)H1)(f) = − 1|xα1 × xα1 |2

D2f +A0(f). (120)

It follows from (119) and (120) that the leading order terms in dδli/dt cancel each

other. We get

dδ1idt

= A0(x) +A0(F ), (121)

dδ2idt

= A0(x) +A0(F ). (122)

As a result, the error equations simplify further to the leading order,

dδ1idt

= A0(x) +A0(F ), (123)

dδ2idt

= A0(x) +A0(F ), (124)

∂xni

∂t= |Dh

1xi ×Dh2xi|Λs

hFi +A0(x) +A0(F ) (125)


12|wi|2, (126)

where Λsh = Hs

1,hDh1 +Hs

2,hDh2 .


Energy estimates and convergence analysis

In the above estimates, we only study the linear stability of the modified pointvortex approximation. To obtain convergence, we also need to obtain nonlinearstability. To this end, we use Strang’s technique [24], which can be summarizedas follows. In proving convergence, there are two steps, consistency and stability.In the consistency step, we usually use the exact solution of the continuum waterwave equations to construct “exact particles” x(αi, t) and vortex sheet strengthsγl(αi, t). Denote by R(t) (the consistency error) the amount by which these exactparticles fail to satisfy the modified point vortex equations. In our case, we haveR(t) = O(h3). Strang’s idea is not to use the exact particles, but to construct“smooth particles” (x(αi, t) = x(αi, t)+h3x3(αi, t), γl(αi, t) = γl(αi, t)+h3γl3(αi, t)for some smooth x3 and γl3) that are O(h3) perturbations of the exact particles,which satisfy the discrete equations more accurately: R(t) = O(hr) for r ≥ 5 aslong as the continuous solution is sufficiently smooth. Existence of such smoothparticles is guaranteed by the existence of the error expansion we obtained for theconsistency error in section 5. The perturbed solution, x3 and γl3, basically satisfiesthe linearized water wave equations with coefficients depending the exact solution ofthe water wave equations. Then, in the stability step, we bound e(t), the differencebetween the smooth particles and the particles computed by the modified pointvortex method. As observed by Strang in [24], if the numerical method is linearlystable, nonlinear stability can be obtained by using the smallness of the error e(t).This greatly simplifies the nonlinear stability analysis.

To illustrate, we define

T ∗ = sup{t | t ≤ T, ‖x‖l2 ≤ h3, ‖γ‖l2 ≤ h2}.

Here the errors x and γ are with respect to the “smooth” particles x(α, t) and thesmooth vortex vortex strengths γ(α, t) [24, 4, 8]. Since h2|xi|2 ≤ ‖x‖2l2 , we concludethat

‖x‖∞ ≤ 1h‖x‖l2 ≤ h2, for t ≤ T ∗ . (127)

Similarly we have

‖γ‖∞ ≤ h, for t ≤ T ∗,

and

‖Dhl x‖∞ ≤ 1

h‖Dh

l x‖l2 ≤2πh2‖x‖l2 ≤ 2πh, l = 1, 2, for t ≤ T ∗.

A typical nonlinear term has the following form:

ENLi =

∑j6=i

γlj|xi − xj|2

|(x(αi)− x(αj)) + (xi − xj)|3h2.

We assume that the exact water wave solution is smooth and non-self-intersecting.Thus we have

|x(αi)− x(αj)| ≥ c|αi − αj| ≥ ch, for j 6= i.

Therefore, for h small and t ≤ T ∗, we have from (127) that

|x(αi)− x(αj)) + (xi − xj)| ≥ c|αi − αj| − 2h2 ≥ c

2|αi − αj|.


Thus, for t ≤ T ∗, we get

‖ENL‖l2 ≤ ‖γ‖∞‖x‖l2maxi

∑j6=i

Ch2

|αi − αj|3≤ C

h‖γ‖∞‖x‖l2 ≤ C‖x‖l2 , (128)

where we have used Young’s inequality. This shows that ENL = A0(x). Othernonlinear terms can be treated similarly.

Combining the consistency and stability steps, we obtain the following new errorequations,

dδ1idt

= A0(x) +A0(F ) +O(hr), (129)

dδ2idt

= A0(x) +A0(F ) +O(hr), (130)

∂xni

∂t= |Dh

1xi ×Dh2xi|Λs

hFi +A0(x) +A0(F ) +O(hr), (131)

dFi

dt= −c(αi)xn

i +12|wi|2, (132)

where we still use the same notation, x, to denote the error between the particlescomputed by the modified point vortex method and the smooth particles x, andr ≥ 5 by construction.

To perform energy estimate, we first define a discrete H1/2 norm. From (78), wehave

(Λf)(α) =∫g2(ξ, α)f(ξ)eiξ·αdξ, (133)

where g(ξ, α) is defined as

g(ξ, α) =1

|xα1 × xα2 |(|xα2 |2ξ21 − 2(xα1 · xα2)ξ1ξ2 + |xα1 |2ξ22)1/4 (134)

Introduce a function ψ(α) defined as follows

ψ(α) =∫g(ξ, α)f(ξ)eiξ·αdξ.

We can show that if f is a real function, so is ψ. To see this, we compute thecomplex conjugate of ψ. We have

ψ(α) =∫g(ξ, α)f(ξ)e−iξ·αdξ =

∫g(ξ, α)\f(−ξ)e−iξ·αdξ (135)

=∫g(−ξ, α)f(ξ)eiξ·αdξ = ψ(α), (136)

where we have used the fact that g is an even function of ξ and f(ξ) =\f(−ξ) sincef is real.

By using the same argument as in the proof of Lemma 3.2 in [18], we can showthat the Λ operator satisfies the following estimate:

(Λf, f)− (ψ,ψ) = (A0(f), f), (137)

where (f, g) is the usual inner product in L2, and A0 is a bounded operator in L2.Furthermore, we assume that |(A0(f), f)| ≤ c0‖f‖L2 . Define a generalized H1/2

norm as follows:

‖f‖2H1/2 = ((Λ + dI)f, f),


where d = c0 + 1. It follows from (137) that

((Λ + dI)f, f) = (Λf, f) + d(f, f)= (ψ,ψ) + (A0(f), f) + d(f, f)≥ (ψ,ψ) + (f, f)≥ 0.

Recall that ψ(α) =∫g(ξ, α)f(ξ)eiξ·αdξ, and |g(ξ, α)| ≤ c|ξ|1/2. Thus we also

have

((Λ + dI)f, f) ≤ C‖f‖2H1/2 .

Since Λsh is a spectral approximation of Λ, we can introduce a discrete H1/2 norm

as follows:

‖f‖H

1/2h

= ((Λsh + dI)f, f)1/2

h , (138)

where (·, ·)h is the inner product associated with the discrete l2 norm. Then theabove derivation for the continuous case can be carried out in the same way for thediscrete norm.

By assumption of the theorem, the problem is well-posed. It has been shown byWu [26] that the coefficient c(α) in (126) is positive

c(α) ≥ c0 > 0.

To obtain our energy estimate, we multiply (129) by δ1i , (130) by δ2i , (131) byc(αi)

|Dh1 xi×Dh

2 xi|xn, (132) by (Λs

h + dI)Fi, we then add and sum in i. Let

y20(t) = ‖

(c(αi)

|Dh1xi ×Dh

2xi|

)1/2

xn(t)‖2l2 + ‖δ1(t)‖2l2 + ‖δ2(t)‖2l2 + ‖F (t)‖2H

1/2h

,

where

‖F (t)‖2H

1/2h

= ((Λsh + dI)F (t), F (t))h.

We obtain12dy2

0

dt= (Λs

hF , cxn)h − (cxn, (Λs

h + dI)F )h

+ (f1, δ1)h + (f2, δ2)h + (f3, xn)h + (f4, (Λsh + dI)F )h, (139)

where

‖fj‖l2 ≤ C(‖xn‖l2 + ‖δ1‖l2 + ‖δ2‖l2 + ‖F‖l2 + hr), j = 1, 2, 3,

and f4 = 12 |wi|2.

Note that wi = dxi

dt . Using the estimate on dxi

dt and the fact that ‖x‖∞ ≤ h2 and‖γ‖∞ ≤ h for t ≤ T ∗, we can show that ‖w‖∞ ≤ c h| log(h)|. Therefore, we obtain

|( (Λsh + dI)F , |wi|2)h| ≤ ‖(Λs

h + dI)F‖l2‖ |w|2‖l2

≤ c h | log(h)| ‖(Λsh + dI)F‖l2

(‖(Λs

h + dI)F‖l2 + ‖F‖l2 + ‖x‖l2 + hr).

It follows from the spectral property of Λsh that

‖(Λsh + dI)F‖l2 ≤ Ch−1/2‖F‖

H1/2h

.


As a consequence, we get

|((Λsh + dI)F , |wi|2)h| ≤ C‖F‖H1/2(y0(t) + chr).

Now we are ready to complete the convergence analysis. Note that the twoleading order terms in the righthand side of (139) containing the Λs

h operator cancelseach other and the entire right-hand side is bounded by y0(t)(y0(t) +Chr). Hencewe obtain

dy20

dt≤ C0 y0(t)(y0(t) + hr),

for some constant C0. The Gronwall inequality then implies

y0(t) ≤ C(T )hr, t ≤ T ∗. (140)

In terms of the original variables, we have

‖x‖l2 ≤ B(T )hr, ‖γ‖l2 ≤ B(T )hr−1, t ≤ T ∗, (141)

where we have used (93) for γ. Since r ≥ 5, for h small enough, we get

‖x‖l2 ≤ B(T )h5 <12h3, ‖γ‖l2 ≤ B(T )h4 <

12h2.

It follows from the definition of T ∗ that

T ∗ = T.

This implies that estimate (141) is valid for the entire time interval 0 ≤ t ≤ T .Since the smooth particles and the smooth vortex sheet strengths are order O(h3)perturbations from the exact particles and vortex sheet strengths, i.e. x(α, t) =x(α, t)+O(h3) and γl(α, t) = γl(α, t)+O(h3) (l = 1, 2) by construction, we concludethat

‖x(t)− x(·, t)‖l2 ≤ Ch3, 0 ≤ t ≤ T, (142)‖γl(t)− γl(·, t)‖l2 ≤ Ch3, 0 ≤ t ≤ T. (143)

This completes the convergence proof of the modified point vortex method for 3-Dwater waves.

Appendix A. Proof of Lemma 7.1

Proof of Lemma 7.1: The invertibility of of I + 2Ah will follow from that of theoperator I + 2A of which it is an approximation, where A is the two by two matrixoperator acting on periodic functions of α. We have discussed the invertibility ofI + 2A, and now we treat Ah as a perturbation of A.

We would like to extend the discrete operator PhrsK

hl (γk)j, r, s, , k = 1, 2, l =

1, 2, 3, 4 to a continuous operator in such a way that the L2 norm of the extendedcontinuous operator is the same as the discrete l2 norm of the discrete operator andthat the extended operator Ph

rs is a bounded operator. We do this in two steps. Inthe first step, we extend fj = Kh

l (γk)j as a piecewise constant function in L2 usingthe same method as in Lemma 6.1 of [4].

In the second step, we will extend Phrsf(α) for periodic function f defined in

[−π, π)2. Keep in mind that we need to define our extension in such a way that theL2 norm of the extended operator should be the same as the discrete l2 norm ofthe original discrete operator. Let Bj = {α ∈ [−π, π)2 : 0 ≤ αl− (αj)l < h, l = 1, 2}


for j ∈ I, where I is the two-dimensional integer set. Define the average of f overBj as follows:

fj =1|Bj|

∫Bj

f(x)dx.

Since f(α) is a piecewise constant function on Bj from our step 1, we have

fj = fj.

Let IN be a two-dimensional integer set defined by

IN = {(j1, j2), | −N/2 + 1 ≤ j1 ≤ N/2, −N/2 + 1 ≤ j2 ≤ N/2}.Using the discrete Fourier transform, we havef(k) = h2

∑j∈IN

fjeik·αj ,

andf(k) = h2

∑k∈IN

fjeik·αj .

Since fj = fj, we have f(k) = f(k). Now we define the extension of Ph12 as follows:

Ph12f(α) =

∑k∈IN

k1k2

|k|2f(k)eik·α.

We claim that

‖Ph12f(α)‖L2 = ‖Ph

12f(αj)‖l2 , (144)

for any piecewise constant function f(α).Proof of (144): Using Parseval equality and the fact f(k) = f(k), we have

‖Ph12f(α)‖2L2 = ‖

∑k∈IN

k1k2

|k|2f(k)eik·α‖2L2

=∑k∈IN

|k1k2

|k|2f(k)|2

=∑k∈IN

|k1k2

|k|2f(k)|2.

On the other hand, using the discrete Parseval equality, we get

‖Ph12f(αj)‖2l2 = ‖

∑k∈IN

k1k2

|k|2f(k)eik·αj‖l2

=∑k∈IN

|k1k2

|k|2f(k)|2.

Thus we have

‖Ph12K

hl (γk)(α)‖L2 = ‖Ph

12Khl (γk)j‖l2 , l = 1, 2, 3, 4.

Similar result applies to other Phrs operators, r, s = 1, 2.

Next we will prove

‖Phrsf(α)‖L2 ≤ ‖f‖L2 , (145)

for any L2 function f(α).


Proof of (145): From the definition of the extension operator and using theParseval equality, we get

‖Ph12f(α)‖2L2 = ‖

∑k∈IN

k1k2

|k|2f(k)eik·α‖L2 =

∑k∈IN

|k1k2

|k|2f(k)|2

≤∑k∈IN

|f(k)|2 = h2∑k∈IN

|fj|2 ≤∑j∈IN

∫Bj

|f(α)|2dα = ‖f‖2L2 .

Using (144)-(145), and arguing as in the proof of Lemma 6.1 in [4], we can showthat

‖A−Ah‖L2h

= O(h| log h|).Finally, it follows from Theorem 1.37 in [7] that (1

2I + Ah +O(h))−1 exists and isbounded since the unperturbed operator ( 1

2I +A)−1 exists and is bounded.

Appendix B. Proof of Khl (φ) = A−1(φ) (l = 1, 2, 3, 4)

Proof of Khl (φ) = A−1(φ) (l = 1, 2, 3, 4).

Similar to the regularized Green’s function Gh in [4], we define

Gh(x) = −(4πr)−1s(r/h), r = |x|,with s chosen so that s(r) is smooth and

s(r) ={

0, r < c/2,1, r > 3c/4,

where the constant c is chosen such that |x(α, t)− x(α′, t)| ≥ c|α− α′|. Therefore,Gh is smooth, for h > 0 and we have Gh(x) = h−1G1(x/h). Letting ψ = ∆G1, wefind

ψ(x) = ∆G1(x) = −(4πr)−1s′′(r), r = |x|,and correspondingly

∆Gh(x) = ψh(x) = h−3ψ(|x|/h),which approximates the Dirac delta function as h −→ 0.

Since 5xGh(0) = 0 by the definition of Gh, we can rewrite Kh1 as follows:

Kh1 (φ) =

∑j6=i

Dh1xi ×Dh

2xj · 5x′G(xi − xj)φjh2

=∑j

Dh1xi ×Dh

2xj · 5x′Gh(xi − xj)φjh2

≡∑j

Kh(xi,xj)φjh2.

The following proof is similar to the proof of Theorem 5.1 in [4]. We first splitthe regularized Green’s function into a far-field part and local part, in order to focusattention on the latter. We wish to restrict the local analysis to a small neighbor-hood of the singularity in which the coordinate mapping is well approximated byits linearization. Let J(α) be the Jacobian matrix ∂x

∂α at α. It can be shown thatthere is some δ0 small enough with δ0 < π/2 so that for all α and α′ satisfying|α− α′| ≤ δ0 we have

|J(α)(α− α′)|/2 ≤ |x(α)− x(α′)| ≤ 2|J(α)(α− α′)|.


It follows from the above inequalities and the fact that the Jacobian matrix J(α) isnonsingular that there exists r0 > 0 with r0 < π/2 so that |x(α)−x(α′)| ≤ r0 implies|α − α′| ≤ δ0. We can also assume that |J(α)(α − α′)| ≤ δ0 implies |α − α′| ≤ δ0.We now choose a cut-off function ζ so that ζ(x) = 1 for x near 0 and ζ(x) = 0 for|x| > r0. We write the regularized Kernel function as Kh = ζKh + (1− ζ)Kh. Thefar-field part K∞

h = (1−ζ)Kh is smooth. Consequently a discrete integral operatorwith kernel DmK∞

h (xi,xj) is l2-bounded, uniformly in h, where Dm is a derivativeoperator of any order m. The local term Kh when evaluated on the surface is

Kh(x(α),x(α′))ζ(x(α)− x(α′))

for |α−α′| ≤ π; the remaining terms in the sum (those corresponding to |α−α′| >π) are zero because the small support of ζ. Thus the boundedness properties ofa discrete operator with DmKh(xi,xj) reduce to consideration of DmK0

h(xi,xj),where K0

h(xi,xj) = Kh(xi,xj)ζ(xi − xj).We now consider the boundedness properties of the operator Kh

1 . Because ofthe above remark, we can replace Kh by K0

h. We will make a Taylor expansion ofthe kernel. We treat xi−xj as a perturbation of its linearization yij = Ji(αi−αj),where Ji = J(αi). Thus we can write DGh(xi − xj) as an expansion in termsDm+1Gh(yij)(zij)m, summed over m with remainder, where zij = x(αi)− x(αj)−yij, and m is a multi-index. We can further expand zij in powers of αi − αj,quadratic or higher. We can also expand D2xj as D2xi plus powers of αi −αj. Wethen have a linear combination of terms for Kh. A typical term for |αi−αj| ≤ π isgiven by

Dm+1x Gh(Ji(αi − αj))(αi − αj)p+l. (146)

Here p, l are some multi-indices, with |p| ≥ 2|m|, |l| ≥ 0. For a discrete integraloperator with (146) as a kernel, we can derive boundedness properties of this opera-tor by estimating the Fourier transform of the kernel αp+lDm

x Gh(Jiα) with respectto α. The relevant estimates have been stated in the lemma 5.3 in [4]. These es-timates and the previous remarks show that the operator with kernel (146) gains|p|+ |l| − |m| derivatives, i.e., it is of order |m| − |p| − |l| in l2 in the sense definedabove. The first term, with m = p = l = 0 vanishes because Dlx(αi) ·N(αi) = 0(l = 1, 2), and the others are of order −1 or less since |p| ≥ 2|m|. This proves thatKh

1 (φ) = A−1(φ). Similar argument applies to Khl for l = 2, 3, 4.

Appendix C. Derivation of (81)-(86)

Estimates for Elpvm (l=1,2)

By the definition of E1pvm(i), we have


E1pvm(i) = Dh

1 xi ·∑j6=i

ηj ×5x′G(xi − xj)h2

+Dh1xi ·

∑j6=i

(γ1jDh2xj − γ2jD

h1xj)×5x′G(xi − xj)h2

+Dh1xi ·

∑j6=i

(γ1jDh2 xj − γ2jD

h1 xj)×5x′G(xi − xj)h2

+Dh1xi ·

∑j6=i

ηj × Er(5x′G(xi − xj))h2

−γ1i

∑j6=i

Nj · 5x′G(xi − xj)h2

−γ1i

∑j6=i

(Dh1 xj ×Dh

2xj +Dh1xj ×Dh

2 xj) · 5x′G(xi − xj)h2

−γ1i

∑j6=i

Nj · Er(5x′G(xi − xj))h2,

where Er(f) = f(αj)− fj. The first term is Dh1 xi ·wh

0 ; the second term is Kh1 γ1 −

Kh2 γ2; the fourth and seventh terms cancel each other to the leading order; the

fifth term gives O(h)γi. Combining the third and sixth terms, we can obtain Ni

2 ·(γ1iH

p1,h + γ2iH

p2,h)Dh

1 xi to the leading order.Similarly, we can derive (83) for E2

pvm(i).

Estimates for Enpvm

By the definition of Enpvm(i), we have

Enpvm(i) = ni ·

∑j6=i

ηj ×5x′G(xi − xj)h2

+ni ·∑j6=i

(γ1jDh2xj − γ2jD

h1xj)×5x′G(xi − xj)h2

+ni ·∑j6=i

(γ1jDh2 xj − γ2jD

h1 xj)×5x′G(xi − xj)h2

+ni ·∑j6=i

ηj × Er(5x′G(xi − xj))h2

−Er( ηi|Ni|

)∑j6=i

5x′G(xi − xj)×Njh2

− ηi|Ni|

∑j6=i

5x′G(xi − xj)× (Dh1 xj ×Dh

2xj +Dh1xj ×Dh

2 xj)h2

− ηi|Ni|

∑j6=i

Er(5x′G(xi − xj))×Njh2,

where we have used the notation Er(f) = f(αj)− fj. The first term is ni ·wh0 ; the

second term is |Ni|2 (Hh

1,p(γ1i)+Hh2,p(γ2i)); the fourth and seventh terms cancel each

other to the leading order; the fifth term gives O(h)γi + O(h)A1(x) = A−1(γi) +A0(x). Combining the third and sixth terms, we obtain to the leading order the


following error terms:


p1,hD

h1 +Hh

2,hDh2 )xi − γ2iD

h2x∗i · (H

p1,hD

h1 +Hh

2,hDh2 )xi.

Putting all the leading order estimates together, we get the desired estimates forEn

pvm.

Estimates for Enear

By the definition of E1near(i), we have

E1near(i) = Er(

Ni

2γ1i) · (Hs

1,h −Hp1,h)Dh

1xi + Er(Ni

2γ2i) · (Hs

2,h −Hp2,h)Dh

1xi

+(Ni

2γ1i) · Er(Hs

1,h −Hp1,h)Dh

1xi + (Ni

2γ2i) · Er(Hs

2,h −Hp2,h)Dh

1xi

(Ni

2γ1i) · (Hs

1,h −Hp1,h)Dh

1 xi + (Ni

2γ2i) · (Hs

2,h −Hp2,h)Dh

1 xi,

where we have used the notation Er(f) = f(αj)− fj. The first term is O(h)(γi) +O(h)A1(xi) = O(h)(γi) + A0(xi); the second term is of the same order; the fifthand sixth terms are the terms that we would like to get. We claim that the thirdand fourth terms are O(h)A1(xi) = A0(xi). Let us study the third term in somedetails. It can be expressed as follows:

(Ni

2γ1i) · (Dh

1xi ·Dh1 xi)(Hs

1,11,h −Hp1,11,h)Dh

1xi

+(Ni

2γ1i) · (Dh

1xi ·Dh2 xi)(Hs

1,12,h −Hp1,12,h)Dh

1xi

+(Ni

2γ1i) · (Dh

2xi ·Dh1 xi)(Hs

1,21,h −Hp1,21,h)Dh

1xi

+(Ni

2γ1i) · (Dh

2xi ·Dh2 xi)(Hs

1,22,h −Hp1,22,h)Dh

1xi

where the operator H1,lm, l,m = 1, 2 is an integral operator with the followingkernel

(α1 − α′1)(αl − α′l)(αm − α′m)|r|5

,

and

r = Dh1xi(α1 − α′1) +Dh

2xi(α2 − α′2).

As a consequence, we can show that the third term is O(h)A1(xi) = A0(xi). Sim-ilarly, we can show that the fourth term is of the same order. This completes thederivation for E1

near. Using a similar argument, we can derive (83) and (84) forE2

near(i) and Ennear(i) respectively.

Acknowledgements: We would like to thank Professor Hector Ceniceros forproof reading the original manuscript and for making several useful comments.The research was in part supported by a grant from National Science Foundationunder the contract DMS-0073916, and by a grant from Army Research Office underthe contract DAAD19-99-1-0141. Pingwen Zhang also wishes to acknowledge thesupport of the Special Funds for Major State Basic Research Projects G1999032804from China.


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Received for publication October 2001.E-mail address: [email protected]

E-mail address: [email protected]

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